The origin of cooperative solubilisation by hydrotropes
Seishi Shimizu1,* and Nobuyuki Matubayasi2,3
1York Structural Biology Laboratory, Department of Chemistry, University of York, Heslington, York YO10 5DD, United Kingdom
2Division of Chemical Engineering, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
3Elements Strategy Initiative for Catalysts and Batteries, Kyoto University, Katsura, Kyoto 615-8520, Japan
KEYWORDS: Hydrotropy; McMillan-Mayer Theory; Water; Cosolvent; Cooperativity; Minimum Hydrotrope Concentration
AUTHOR INFORMATION
Corresponding Author:
Seishi Shimizu
York Structural Biology Laboratory, Department of Chemistry, University of York, Heslington, York YO10 5DD, United Kingdom
Tel: +44 1904 328281, Fax: +44 1904 328281, Email:
ABSTRACT
The signature of hydrotropic solubilisation is the sigmoidal solubility curve; when plotted against hydrotrope concentration, solubility increases suddenly after the minimum hydrotrope concentration (MHC), and reaches a plateau at higher hydrotrope concentrations. This sigmoidal curve is characteristic of cooperative phenomena, yet the true molecular basis of hydrotropic cooperativity has long remained unclear. Here we develop a theory, derived from the first principles of statistical thermodynamics using partially-open ensembles, to identify the origin of hydrophobic cooperativity. Our theory bears a close resemblance to the cooperative binding model used for protein-ligand binding. The cause of cooperativity is the enhancement of hydrotrope m-body interaction induced by the presence of the solute; m can be estimated from experimental solubility data.
1. Introduction
Hydrotropes can increase the solubility of hydrophobic solutes up to several orders of magnitude, hence have a number of important industrial applications.1-6 The signature characteristics of hydrotropes, which sets them apart from other cosolvents, is the sigmoidal solubility curve, or more specifically:2-6
1. Solubility hardly increases at low hydrotrope concentrations (<0.5 molar).
2. Above a certain threshold concentration, commonly referred to as the minimum hydrotrope concentration (MHC), solubility increases suddenly.
3. Solubility ceases to increase after a few molars of hydrotropes (saturation of solubilisation).
Such sigmoidal solubilisation curves (Figure 1) are reminiscent of cooperative phenomena.7
What is the origin of hydrotropic cooperativity? Because of the apparent similarity between MHC and critical micelle concentration (CMC), 7-9 the self-aggregation of hydrotrope in the bulk phase has often been considered to be the origin of hydrotropic cooperativity.7-13 However, the fact that MHC is observed even for urea, which forms near-ideal mixture with water, has seriously challenged this hypothesis.14-17
Rigorous statistical thermodynamics, on the contrary, has shown that bulk-phase self-aggregation is not the cause of MHC.14-17 Furthermore, such bulk-phase self-aggregation reduces the effective number of hydrotrope molecules, thereby reducing the solubilisation efficiency per hydrotrope molecule.14-17 Instead, MHC is caused by the enhanced hydrotrope self-aggregation in the presence of the solute.16 This conclusion was reached through the use of the rigorous Kirkwood-Buff (KB) theory,18-30 which identified, without any approximations, the driving forces of hydrotropic solubilisation,14-17 which has later been supported by further evidence.6,31
Even though our rigorous KB-based approach has revealed the new, universal principles of hydrotropy, there still are two shortcomings: (i) the mathematical form of the theory on the origin of MHC is complex and is difficult to use;14-17,32 (ii) the origin of solubilisation plateau remains unexplained.16,17
Hence there is a need for a simple theory, which can identify the cause of hydrotropic cooperativity. We start from the first principles of statistical thermodynamics.30,33,34 Guided by an analogy between ligand binding cooperativity35-38 and hydrotropic cooperativity, we propose how solubility curve should be analysed, in order to reveal the key aggregation interactions responsible for hydrotropic cooperativity.
2. A statistical thermodynamic foundation for hydrotropic cooperativity
The goal of our theory is to express solubility as a function of hydrotrope concentration. We first note that solubility measurements are usually carried out in the isobaric-isothermal (NPT) ensemble, whereas the concentration polynomial expansion of thermodynamic quantities is carried out in the grand canonical (μVT) ensemble.30,32-34 Our goal is to obtain such an expansion under an isobaric ensemble. This can be done by the use of the partially-open isobaric ensemble pioneered decades ago by Stockmeyer and Hill.39-43
Consider a three component solution consisting of a solute (i=u), water (i=1), and hydrotrope (i=2) molecules. Let μi and Ni respectively be the chemical potential and the number of species i, T, V and P respectively be the temperature, volume, and pressure of the system. Let ρi=Ni/V be the number density or concentration of species i; when Ni is not kept constant, the ensemble average of Ni is used instead to define ρi, such that ρi=Ni/V. The convention β=1kT (where k is the Boltzmann constant) is used throughout. Since T is kept constant throughout this paper, it is often omitted in the subsequent discussions.
To calculate the solvation free energy of a solute molecule, the pair of systems, with and without the solute, needs to be considered.16 When the solute is present, it is fixed at the origin and acts as the source for an external field for the water and hydrotrope molecules. In this case, the solution system is inhomogeneous.16 When the solute is absent, the system consists only of water and hydrotrope; hence this system is homogeneous. The partially-open partition functions for the two-component system with and without the solute molecule can be expressed respectively in the following16,30,32-34
ΓuT,P,N1,μ2=N2≥0λ2N2RuT,P,N1,N2 (1)
ΓT,P,N1,μ2=N2≥0λ2N2RT,P,N1,N2 (2)
where λi is the fugacity of the species i, is defined as16,30,32-34
λi=expβμi (3)
and the isobaric-isothermal partition functions are defined as
RuT,P,N1,N2=1N1!N2!q1N1q2N2Λ13N1Λ23N2dV e-βPVdxudXN1dXN2e-βU(xu, XN1,XN2) (4)
RT,P,N1,N2=1N1!N2!q1N1q2N2Λ13N1Λ23N2dVe-βPVdXN1dXN2e-βU(XN1,XN2) (5)
where Λi is the de Broglie wavelength, qi is the intramolecular partition function, XN1 and XN2 denote collectively the coordinates of the species 1 and 2, respectively, and xu is the internal coordinates of the solute.
Connection to thermodynamics can be made through the following formulae
N1μ1u=-kTlnΓuT,P,N1,μ2 (6)
N1μ1=-kTlnΓT,P,N1,μ2 (7)
in which T, kept constant throughout the discussion, is omitted. Most importantly, the chemical potential of the fixed solute μu* can be expressed in terms of the partially-open partition functions in the following manner:16,32
e-βμu*=e-βN1μ1u-N1μ1=ΓuT,P,N1,μ2Γ(T,P,N1,μ2) (8)
The main concern of this paper is the solubility increase which accompanies the introduction of the hydrotrope molecule. This, in the language of thermodynamics, is due to the change of μu* from its pure water value μu*0
Δμu*=μu*-μu*0 (9)
Solubility increase is expressed, using eqn (9), as follows
e-βΔμu*=ΓuT,P,N1,μ2Γ(T,P,N1,μ2)ΓT,P,N1∞Γu(T,P,N1,∞)=ΓuT,P,N1,μ2Γu(T,P,N1,∞)ΓT,P,N1,μ2ΓT,P,N1∞ (10)
where Γu(T,P,N1,∞) and ΓT,P,N1∞ refers to the partition functions in pure-water solvent, in which the concentration of the hydrotrope is 0 and its chemical potential μ2 diverges.
3. The local subsystem open only to hydrotropes
To construct a theory of hydrotropy in an analogous manner to the cooperative binding theory, here we aim to express e-βΔμu* in terms of the “local” distribution of hydrotropes around the solute. To this end, let us introduce a “local” subsystem around the solute, which lies within the macroscopic systems introduced in Section 2. Let the boundary of this inhomogeneous subsystem be the range of solute-hydrotrope correlation, and v be the volume of this subsystem, which is kept constant throughout. Following the classical works of Stockmeyer37 and Schellman,38 let this subsystem be partially open, namely open only to the hydrotropes. We shall later show how to specify v from the behaviours of Ru and R.
Let us now see the consequence of introducing the local subsystem. To this end, let us first note that there are N2!n2!N2-n2! ways of choosing n2 molecules out of N2 identical molecules to be placed within the local subsystem. Using the constancy of v in order to move dXn2 out of the total volume integral dV in the isobaric ensemble, the partially-open partition function can be rewritten as
ΓuT,P,N1,μ2=N2≥0λ2N2n2=0N2N2!n2!N2-n2!RuT,P,N1,N2
=n2≥0λ2n2n2!q2n2Λ23n2dXn2RuT,P,N1,N2b;Xn2ΓuT,P,N1,∞ (11)
where, at the last step, a new variable is introduced as N2b=N2-n2. Note that the kernel of the integration has been denoted here as RuT,P,N1,N2b;Xn2ΓuT,P,N1,∞, because of a clear physical meaning which can be attributed to RuT,P,N1,N2b;Xn2. This can be appreciated by rewriting, using eqn (11), the numerator of eqn (10) into the following form:
ΓuT,P,N1,μ2Γu(T,P,N1∞)=n2≥0λ2n2Ru,n2 (12)
Ru,n2=q2n2n2!Λ23n2dXn2RuT,P,N1,N2b;Xn2 (13)
RuT,P,N1,N2b;Xn2, according to eqn (13), has a clear physical meaning: the fugacity of inserting n2 identical hydrotrope molecules fixed at the configuration Xn2.
Importantly, Ru,n2 is a microscopic quantity, since the range of the integral is over the local system, which is microscopic. In a similar vein, a homogeneous subsystem must be defined in order to complete the link between solubilisation and local hydrotrope distribution. Let us consider the same volume v, which does not contain any solute molecules and sets its origin at the centre-of-mass position of the solute that is to be inserted.
Using the same argument which led to eqn (12) and (13), the following relationships for the bulk solution:
ΓT,P,N1,μ2Γ(T,P,N1∞)=n2≥0λ2n2Rn2 (14)
Rn2=q2n2n2!Λ23n2dXn2RT,P,N1,N2b;Xn2 (15)
Solubilisation can now be linked to the local distribution of hydrotropes; this can be achieved by combining eqn (10), (12)-(15) in following form:
e-βΔμu*=1+n2≥1Ru,n2λ2n21+n2≥1Rn2λ2n2 (16)
We emphasise that Ru,n2 and Rn2 in eqn (16) are microscopic.
Now we derive a rational polynomial expansion of e-βΔμu* based upon eqn (16). To do so, we employ the elegant method of the MM theory, namely to consider N2b→0 limit of Ru,n2 and Rn2. In our definition of the local subsystem, v represents the range of correlation between solute and hydrotrope-hydrotrope interaction; putting N2b→0 is equivalent to ignoring the contribution from the hydrotrope molecules outside of the correlation range. Hence eqn (16), when considered under the N2b→0 limit, serves as the basis for the rational polynomial expansion of solubilisation that we sought after.
4. Hydrotropic cooperativity versus binding cooperativity
Our result, eqn (16) (at N2b→0 limit) is mathematically analogous to binding polynomials35-37 in the theory of cooperative binding. This can be better appreciated by a trivial rewriting of eqn (16),
e-βΔμu*-1=n2≥1ΔRn2λ2n21+n2≥1Rn2λ2n2 (17)
in which ΔRn2=Ru,n2-Rn2. This equation is analogous to binding polynomials.
The analogy between hydrotropic solubilisation and cooperative binding theories opens up a new possibility towards revealing the mechanism of hydrotropic cooperativity. (In fact it is considered to be the continuation of the classical attempts to extend the theory of binding to weak-nonspecific interactions characteristic of solvation).39,44,45 We assume that the summations in the denominator and numerator of eqn (16) are dominated by a few terms for both, which will greatly simplify the analysis. To this end, let us rewrite eqn (17) in the following manner:
e-βΔμu*-1=n2pn2 Ru,n2-Rn2Rn2 (18)
where
pn2=Rn2λ2n21+∑Rn2λ2n2 (19)
signifies the probability of finding n2 hydrotrope molecules in the local subsystem without the solute. Now, the denominator and numerator of eqn (16) can be shown to be dominated by a few terms if the following conditions are met, regardless of the n2-dependence of pn2:
a. Ru,n2-Rn2Rn2=0 for n2n2-Δn2
b. Ru,n2-Rn2Rn2 is large and positive for n2-Δn2n2n2+Δn2
c. Ru,n2-Rn2Rn2=0 for n2n2+Δn2
d. Δn2 is at its minimum, or Δn2≃0
In Appendix, we show that these four conditions will lead to a unique determination of v and n2, and that v signifies the range within which hydrotrope-hydrotrope interactions are affected by the presence of a solute.
There is also a logically possible yet less likely scenario that pn2 is non-zero only between n2=n2-Δn2 and n2+Δn2, and Ru,n2-Rn2Rn2 is positive in the same range. In either case, if we neglect the peak width such that Δn2=0, eqn (18) can be rewritten as
e-βΔμu,NPT*-1=ΔRn2λ2n21+Rn2λ2n2 (20)
Eqn (20) is analogous to the Hill model of binding cooperativity.35-37,45 Now we modify eqn. (20) in two different ways. The first is completely general; we rewrite eqn (20) in terms of the hydrotrope activity, a2. This can be done by using the the standard chemical potential of hydrotrope μ2o=μ2-RTlna2 and the corresponding fugacity of pure hydrotrope λ2o=expβμ2o, as
e-βΔμu*-1=ΔRn2'a2n21+Rn2'a2n2 (21)
Where
ΔRn2'=λ2on2Ru,n2-λ2on2Rn2 (22)
Rn2'=λ2on2Rn2 (23)
A simplification of eqn (21)-(23) is possible when the aqueous hydrotrope solution obeys the dilute ideal solution. In this case, eqn (20) can be rewritten in terms of the mole fraction of the hydrotrope, x2. This can be achieved by using the dilute ideal standard chemical potential μ2ox=μ2-RTlnx2 and the corresponding standard fugacity λ2ox=expβμ2ox, as
e-βΔμu*-1=ΔRn2''x2n21+Rn2'' x2n2 (24)
where
ΔRn2''=λ2oxn2Ru,n2-λ2oxn2Rn2 (25)
Rn2''=λ2oxn2Rn2 (26)
Note that n2 does not depend on the choice of the standard state of the chemical potential, i.e., n2 is the same for eqn (20), (21) and (24).
We emphasize there that we have introduced the conditions a-d to specify when hydrotrope-induced solubilization behaves in a cooperative manner (eqn (20)). In our present theoretical formalism, the validity of these conditions can only be verified through how well eqn (20), (21) or (24) can fit the solubility data, as will be demonstrated in the next section.
5. Linearised plot for analysing solubility data
In the study of ligand binding, key parameters for binding cooperativity can be obtained from experimental binding data through the linear plot, which can visually show how well the experimental data fits the model.35-37 A formal analogy between hydrotropy and cooperative binding suggests that this powerful method can be extended to hydrotropic cooperativity.
We aim to reproduce the overall shape of the solubility curve by eqn (21) or eqn (24). Here we focus on eqn (24), because aqueous hydrotrope solutions can often be treated as dilute ideal solutions. Eqn (24) contains only three parameters, n2, Rn2'', and ΔRn2''=λ2oxn2Ru,n2-λ2oxn2Rn2 . For simplicity, let us first exploit that solubilisation e-βΔμu* converges to a plateau, whose value will be referred to as e-βΔμu*,sat at large x2, which, according to eqn (24), leads to
Ru,n2Rn2=e-βΔμu*,sat (27)
Combining eqn (24) and (27), we obtain
ln1-e-βΔμu*e-βΔμu*-e-βΔμu*,sat=n2 lnx2+lnRn2'' (28)
which suggests that by plotting ln1-e-βΔμu*e-βΔμu*-e-βΔμu*,sat against lnx2, n2 can be determined from the gradient and lnRn2'' from the intercept.
6. Hydrotropic cooperativity from solubility data
Now we apply eqn (28) to analyse experimental data.15,47-49 As model systems, the solutes butyl acetate (BA) and benzyl benzoate (BB) in water are solubilized by the hydrotropes urea, sodium benzoate (sb) and sodium salicylate (ss). Solubilization reaches plateau at approximately x2≃0.04 for urea and 0.08 for ss and sb, during which the activity coefficients of water, γ1, hardly deviate from 1, meaning that the aqueous hydrotrope solution can be considered as ideal dilute solution. Hence we use the theory for dilute ideal solutions, eqn (24).